Duchess is the one to ask, but I am fairly sure you can still register them if their breeding is Appaloosa, colored or not. There is some crackdown in showing them alongside colored horses, I believe. Many solid appearing mares make nice color-producing broodmares.
I think it's like Paints. You can register Paints with no color if they are Paint by breeding... they are called 'Breeding Stock Paints". Depending on conformation, they can be nearly as valuable as colored horses.
If the horse is purebred, they are registerable in the breed. Appaloosa is not just a color registry, and, any more, neither is Paint. It's a breed that has preferred markings.
Retaining the motivation to produce color is always important in breeds where color is unique, and so there is often debate and limitations placed on showing non-colored stock.
If it does not have appaloosa characteristics (striped hooves, white sclera around the eye, mottled skin and/or a coat pattern) it is registered as NC and can be used as breeding stock only - not to be shown in ApHC shows. You can get around this by having the horse CPO'd (Certified Pedigree) if both parents are registered apps.
Unlike Paints or Palominos - an appaloosa is not a color breed, it is a blood breed, although color is a characteristic of the breed. In other words, a solid colored app is as much an appaloosa as is a loud colored appaloosa.